Nonsplitting of the Hilbert exact sequence and the principal Chebotarev density theorem

Abstract

Let K/k be a finite Galois extension of number fields, and let H K be the Hilbert class field of K. We find a way to verify the nonsplitting of the short exact sequenceby finite calculation. Our method is based on the study of the principal version of the Chebotarev density theorem, which represents the density of the prime ideals of k that factor into the product of principal prime ideals in K. We also find explicit equations to express the principal density in terms of the invariants of K/k. In particular, we prove that the group structure of the ideal class group of K can be determined by reading the principal densities.